#464175

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1769 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,4)(6,7)(9,10)(11,20)(12,25)(13,18)(14,30)(15,19)(16,22)(17,28)(21,38)(23,32)(24,37)(26,31)(27,35)(29,34)(33,39)(36,40);(5,6,8,7)(9,11,15,24,13,22,14,23)(10,16,19,32,18,20,30,37)(12,26,29,39,27,40,28,21)(17,33,25,36,34,38,35,31);(6,7)(9,12)(10,17)(11,21)(13,27)(14,29)(15,28)(16,31)(18,34)(19,35)(20,36)(22,39)(23,26)(24,40)(25,30)(32,38)(33,37);(2,4)(5,7,8,6)(12,27)(17,34)(21,39)(25,35)(26,40)(28,29)(31,36)(33,38)

Polycyclic presentation:

2^9|b^2=g,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371371);